Toolbox
June 20, 2025
A decomposition of a topological space into simple pieces.
\[ V_u = \{x \in \mathbb{R}^d \mid \|x - u\| \le \|x - v\|, \forall v \in S\} \]
\[ \text{Čech}(r) = \{ \sigma \subseteq S \mid \bigcap_{x \in \sigma} B_x(r) \neq \emptyset \} \]
\[ \text{VR}(r) = \{ \sigma \subseteq S \mid \text{diam}(\sigma) \le 2r \} \]
\[ \text{Alpha}(r) \subseteq \text{Čech}(r) \subseteq \text{Vietoris-Rips}(r) \]
A mathematical formalism for quantitatively describing the connectivity of a space, particularly its “holes”.
Real-valued functions used to analyze the topology of manifolds.
Measures the scale or resolution of topological features within data, particularly useful for handling noise.
Concerns how persistence diagrams (and thus the measured features) change when the input function or data undergoes small perturbations.
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Thomas Reinke